# Digamma Function of One Half

## Theorem

$\displaystyle \map \psi {\frac 1 2} = -\gamma - 2 \ln 2$

where:

$\psi$ denotes the digamma function
$\gamma$ denotes the Euler-Mascheroni constant.

## Proof

 $\displaystyle \map \psi {\frac 1 2}$ $=$ $\displaystyle -\gamma - \ln 4 - \frac \pi 2 \map \cot {\frac 1 2 \pi} + 2 \sum_{n \mathop = 1}^0 \map \cos {\frac {\pi n} 2} \map \ln {\map \sin {\frac {\pi n} 2} }$ Gauss's Digamma Theorem $\displaystyle$ $=$ $\displaystyle -\gamma - \ln 4$ Cotangent of Half-Integer Multiple of Pi, noting also that the summation is an empty sum $\displaystyle$ $=$ $\displaystyle -\gamma - 2 \ln 2$ Logarithm of Power

$\blacksquare$